Logarithm Calculator

Logarithms turn huge ranges into manageable numbers — that's why earthquake magnitudes, sound levels in decibels and acidity (pH) use them. This log calculator finds the logarithm of any positive number in any base, shows the common (base 10), natural (base e) and binary (base 2) logs together, and the antilog (the base raised to the number).

How to use it

Enter the number x (must be positive).
Enter the base — 10, 2, e or any positive number except 1.

Log rules

log(ab) = log a + log b · log(a/b) = log a − log b · log(aⁿ) = n log a

Example

An earthquake of magnitude 7 releases about 10^1.5 ≈ 32 times the energy of a magnitude 6. And a ₹1 lakh investment doubling at 12% a year takes log(2) ÷ log(1.12) ≈ 6.1 years — the reason behind the “rule of 72”. For powers, use the exponent calculator, or thescientific calculator for longer expressions.

Frequently asked questions

What is a logarithm?

The logarithm answers “to what power must I raise the base to get this number?”. log₁₀(1000) = 3 because 10³ = 1000, and log₂(8) = 3 because 2³ = 8.

What is the difference between log and ln?

log usually means base 10 (common logarithm), used in pH, decibels and the Richter scale. ln is the natural logarithm, base e ≈ 2.71828, used in growth and decay, calculus and finance.

How do I change the base of a logarithm?

Use the change-of-base rule: log_b(x) = ln(x) ÷ ln(b) = log(x) ÷ log(b). The calculator applies this for any base.

Why can’t I take the log of zero or a negative number?

No power of a positive base gives zero or a negative number, so logarithms are only defined for positive numbers (and a positive base other than 1).